# Questions tagged [lie-superalgebras]

The lie-superalgebras tag has no usage guidance.

79
questions

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### smooth super scheme which is not smooth

I am following the very nice "Notes on fundamental algebraic supergeometry. Hilbert and Picard superschemes" by Bruzzo, Ruiperez and Polishchuk. I am having some problem in order to give ...

1
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0
answers

107
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### symmetrization and invariants

Let $V$ be a supervector space over $\mathbb C$ and let $T^n(V):=V \otimes V \otimes \cdots \otimes V$ and let $S^n(V)$ be the super vectorspace of symmetric tensors. Then we have a cannonical ...

0
votes

0
answers

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### Description of the Lie superalgebra $D(2,1;\alpha)$ as the Lie superalgebra preserving some structure

I'd like to know a description of the Lie superalgebra $D(2,1;\alpha)$ as the Lie superalgebra preserving some structure. For example, the orthosymplectic Lie superalgebra $\mathfrak{osp}(m|2n)$ is ...

4
votes

1
answer

82
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### Real forms of the general linear Lie superalgebra

I'm interested in a classification of the real forms of the general linear Lie superalgebra $\mathfrak{gl}_{m|m}(\mathbb{C})$.
The real forms of the simple complex Lie superalgebras were classified by ...

1
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0
answers

22
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### Do the Tits-Kantor-Koecher construction extend to non unital Jordan algebras?

I have an infinite dimensional Jordan algebra which is not-unital. I would like to study a Lie algebra related to it (if it exists). Natural choice would be to look at the KKT construction, but ...

4
votes

0
answers

115
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### What is the Lie superalgebra generated by permutations?

Consider the group algebra of the symmetric group $\mathbb{C}S_n$. Then there is a corresponding Lie algebra $\mathfrak{L}(S_n)$ defined by
$$[\sigma, \tau] = \sigma\circ\tau - \tau\circ\sigma,$$
...

1
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0
answers

167
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### Orthosymplectic superalgebra

Let $V=V_0 \oplus V_1$ be a $\mathbb Z_2$-graded vector space over $\mathbb C$. Suppose $V$ has an even non-degenerate bilinear form $(-, -)$
which is symmetric on $V_0$, skew symmetric on $V_1$, and ...

2
votes

1
answer

222
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### Primitive elements in the universal enveloping algebra of Lie superalgebra

Let $\mathfrak{g}$ be a Lie superalgebra over $\mathbb{C}$. Denote by $U(\mathfrak{g})$ the universal enveloping algebra of $\mathfrak{g}$. We know that there is a natural super Hopf algebra structure ...

2
votes

0
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89
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### Invariants of Lie superalgebras

Let $V=V_0 \oplus V_1$ be a $\mathbb Z_2$-graded vector space over $\mathbb C$. Suppose $V$ has an even non-degenerate bilinear form $(-, -)$
which is symmetric on $V_0$, skew symmetric on $V_1$, and ...

2
votes

0
answers

52
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### Finding the (1,1) component of $e^{-\mathbb{A}^2}$ for $\mathbb{A}$ a superconnection

Let $E=E^+\oplus E^-$be a holomorphic superbundle over a compact Kahler manifold, and $v:E^+\oplus E^-$ an odd bundle map. Assume that both $E^+$ and $E^-$ are endowed with Hermitian metrics, and ...

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108
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### What does “$d$-generated algebra” mean? [closed]

I would like to know what does ”$d$-generated algebra” mean? In particular “two-generated algebra”?
@KonstantinosKanakoglou @LSpice
Here is an example of a paper concerning “one-generated algebras”: ...

7
votes

1
answer

248
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### Semisimple super Lie algebras

There is a classification of simple Lie algebras in $\text{Vec}_{\mathbb{C}}$ given by Dynkin diagrams. We then have 4 families of simple lie algebras, plus some exceptional ones.
Question: How about ...

6
votes

2
answers

300
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### Lie powers of a graded vector space and Klyachko's theorem

Let $V$ be a $\mathbb{Z}_2$-graded vector space (aka super vector space) and $L(V)$ be the free $\mathbb{Z}_2$-graded Lie algebra (aka super Lie algebra). The free super Lie algebra is also graded by ...

6
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0
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### Reference request: superconformal algebras and representations

I am looking for a book/monograph which deals with superconformal (vertex operator) algebras and their representation theory. What are some good books to understand to begin with the definition of a ...

3
votes

1
answer

198
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### Supersymmetry charge $Q$ as anti-linear and anti-unitary operator

We know the supersymmetry (SUSY) charge $Q$ satisfies the following relation respect to fermion parity operator $(-1)^F$:
$$
(-1)^F Q + Q (-1)^F :=\{Q, (-1)^F \} =0
$$
which defines the anti-...

2
votes

1
answer

135
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### Notation on supergeometry — parity

I know that given a manifold $M$ and its corresponding tangent bundle $TM$ we can call $\Pi TM$ the space of forms parametrized (via charts) by $\{x_i\}_{i=1,\dotsc,n}$ and its corresponding cotangent ...

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2
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247
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### Schur Weyl duality for the supergroup $\text{GL}(m|n)$

Let $G$ be the supergroup $\text{GL}(m|n)$. It has a tautological representation $V= \mathbb{C}^{m|n}$.
For every natural number $d$ we have a natural map $$\Phi_d:\mathbb{C} S_d\to \text{End}_G(V^{\...

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0
answers

40
views

### is this the correct universal property of free Lie superalgebras?

Consider a $\Bbb Z_2$ graded set $A$.
Universal property of free Lie superalgebra $FLS(A)$: Let $\mathfrak g$ be a Lie superalgebra and let $\Phi: A \to \mathfrak g$ be a set map which preserves the $\...

1
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0
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52
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### Poisson reduction in odd/graded Poisson geometry?

I would like to know whether there is any literature on Poisson reduction of $\mathbb Z$- or $\mathbb Z_2$-graded Poisson algebras.
A $\mathbb Z$-graded Poisson algebra with degree $p\in\mathbb Z$ ...

1
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2
answers

338
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### Doubt in the Serre relation and the odd/even roots of a Lie superalgebra

Let $I = \{1,2,\dots,n\}$ and $S \subset I$. The set $I$ will be indexing the simple roots and $S$ will be indexing the odd generators of a Lie superalgebra.
A real matrix $A=(a_{ij})_{i,j\in I}$ is ...

2
votes

1
answer

83
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### Sufficient conditions for unitarity of a representation of a Lie Superalgebra

Suppose we have a Lie superalgebra with triangular decomposition:
\begin{equation}
\mathfrak{g} = \mathfrak{g}^{+} \oplus \mathfrak{g}^{0} \oplus \mathfrak{g}^{-}
\end{equation}
I've seen it stated ...

1
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0
answers

76
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### Quadratic Lie superalgebras

Let $(L,.)$ be a Lie superalgebra endowed with an even supersymmetric non-degenerate and invariant bilinear form $B$ (i.e $(L,.,B)$ is a quadratic Lie superalgebra). If we have the equality $B(x,y.z)=(...

7
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115
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### Infinitesimal description of homogeneous supermanifolds

Lie's third theorem says that if $\mathfrak{g}$ is a real, finite-dimensional Lie algebra, then there is a unique (up to isomorphism) simply-connected Lie group $G$ whose tangent Lie algebra is ...

10
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3
answers

947
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### Hopf structure on the universal enveloping of a super Lie algebra

The universal enveloping algebra of a Lie algebra has a canonically defined Hopf algebra structure. Is the same true of the universal enveloping of a super Lie algebra? A presentation in terms of the ...

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0
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40
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### Cartan integers for Lie superalgebras

Let $\mathfrak g$ be a basic classical simple Lie superalgebra (BCSLSA in short) with Cartan matrix $A$ of some simple system $\Pi$. Then $A$ satisfies the conditions that
1) $\frac{2a_{ij}}{a_{ii}} \...

2
votes

0
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63
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### Super characters in the theory of Lie superalgebras

Weyl character formula for the finite dimensional complex semisimple Lie algebras plays a crucial role in the theory of highest weight modules, where for the highest weight module $V(\lambda)$ we ...

2
votes

1
answer

271
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### Category of typical representations for Lie superalgebras

I am interested in studying the category of typical representations over basic classical simple Lie superalgebras. In particular, I want to know
1) is this category semisimple (character determines ...

2
votes

0
answers

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### Denominator identity for Lie superalgebras

Let $\mathfrak g$ be a basic classic simple Lie superalgebra.
Fix a maximal isotropic subset $S \subset \Delta$ and choose a set of simple roots $\Pi$ containing $S$. Let $R$ be the Weyl ...

2
votes

0
answers

61
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### Dynkin diagram of Basic classical simple Lie superalgebras

Let $\mathfrak g = \mathfrak g_0 \oplus \mathfrak g_1$ be a basic classical simple Lie superalgebra with the root system $\Delta = \Delta_0 \cup \Delta_1$ and Dynkin diagram $\Gamma$. It is well-known ...

4
votes

1
answer

279
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### Typical and atypical modules for Lie superalgebras

There are two types highest weight representations for a Basic classical simple Lie superalgebra $\mathfrak{g}$ which are defined as typical (representation for which highest weight vector is the only ...

4
votes

2
answers

352
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### Lie super algebra presentation of the Kähler identities

For any Kähler manifold $(M,h)$, with Lefschetz operators $L$ and $\Lambda$, and counting operator $H$, we have the following the well-known Kähler-Hodge identities:
\begin{align*}
[\partial,L] = 0, ...

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0
answers

76
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### GKO construction for (Super-)Virasoro algebras

I am reading the paper "Unitary Representations of the Virasoro and Super-Virasoro Algebras" by Goddard, Kent, Olive. In many places, the authors claim results without any justification, or with ...

2
votes

1
answer

134
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### super Lyndon words

Lyndon words form a basis for Free Lie algebras.
In this direction, I need the reference for the super Lyndon words for free Lie superalgebras.
Given the definition of super Lyndon words, how to ...

2
votes

1
answer

119
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### Lyndon basis of free Lie superalgebras

Lyndon basis for Free Lie algebras is well known in the literature.
My question is,
what is the analogous combinatorial model for the case of free Lie superalgebras? what is the super analogous of ...

4
votes

0
answers

67
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### Reference Request: Representation Theory of Real Lie Superalgebras

Are there some references for the representation theory real lie superalgebras, specifically of $psu(1,1|2)$, and $u(1,1|2)$?

5
votes

2
answers

566
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### Serre relations for Lie Superalgebras

Finite dimensional complex simple Lie algebras are classified using Cartan matrices. One of the main ingredients is Serre Relations. Lets call this Cartan-Killing theory.
I have the following ...

2
votes

0
answers

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### Coordinate ring of an equivariant embedding of a homogeneous projective variety

Lie algebra: Let $G$ be a semisimple, simply connected linear algebraic group with a fixed Borel subgroup $B$. Let $P$ be a parabolic subgroup containing $B$. Let $\lambda$ be a dominant integral ...

10
votes

1
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528
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### Kazhdan-Lusztig equivalence for Lie super-algebras

Let $\mathfrak g$ be a semi-simple Lie algebra. Kazhdan and Lusztig studied the category of representations of the corresponding affine Lie algebra (the central extension of $\mathfrak g((t))$) which ...

2
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0
answers

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### odd positive roots of basic classical Lie superalgebras

In the Lie algebra case, positive roots are "almost ($S_{\alpha}$ permutes $\Delta^+ -\{\alpha\}$)" invariant under simple reflections. A similar statement I want to understand for Lie superalgebras.
...

2
votes

0
answers

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### action of Weyl group element on Weyl vector

Let $\mathfrak g = \mathfrak g_0 \oplus \mathfrak g_1$ be a basic classical Lie super algebra and let $\rho = \text{half sum of even positive roots} - \text{half sum of odd positive roots}$ be the ...

1
vote

1
answer

155
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### finite dimensional modules are highest weight modules [closed]

Let $\mathfrak{g}$ be a basic classical simple Lie super algebra. I want to prove that every finite dimensional module over $\mathfrak{g}$ has a highest weight vector.
My feeling is, since $e_i$'s ...

1
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0
answers

73
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### how the borel subalgebras of p(n) look like except the standard one?

I am reading the book Musson: Lie superalgebras. In Chapter 3, on page 62, Lemma 3.6.8 tells about support of Borel subalgebra. I am confused about this.
I am trying to do the proof of the Proposition ...

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votes

2
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658
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### Character formula for Lie superalgebras

The Weyl character formula and the denominator identity play important roles in the representation theory of classical simple Lie algebras and Kac-Moody Lie algebras over $\mathbb{C}$
Can you suggest ...

2
votes

1
answer

129
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### $P(1)$ strange type classical Lie superalgebras

I am reading the book "Lie superalgebras and enveloping algebras" by Ian M. Musson.
The strange type $P(n)$ series of Lie superalgebras are defined (§2.4.1, p. 17) only for $n \ge 2$ even though for $...

0
votes

1
answer

138
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### left ideals in Lie super algebras

Let $\mathfrak g$ be a Lie superalgebra.
If $\mathfrak a$ is not a grade subspace of $\mathfrak g$, then why does $[\mathfrak g, \mathfrak a]$ and $[\mathfrak a, \mathfrak g]$ are not same?
For me ...

2
votes

3
answers

442
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### Graph of a Lie super algebra

Let $A$ be a generalized Cartan matrix and let $\mathfrak{g}$ be the Kac-Moody Lie algebra associated to $A$. There is an associated graph of $\mathfrak{g}$ which is known as the Dynkin diagram of $\...

2
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0
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### Two definitions of super-Virasoro algebra

Let $A=\mathbb C[x,\epsilon]$ where $x$ is an even variable and $\epsilon$ is an odd variable (thus $A$ is a commutative super-algebra). Let $\mathfrak g$ denote the Lie super-algebra of vector fields ...

8
votes

1
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### Vanishing of H-cohomology

This looks elementary, but somehow I am stuck, please bear with me:
Let $H$ be a differential 3-form, nowhere vanishing, but not necessarily closed. What is a sufficient condition that the sequence ...

2
votes

1
answer

276
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### Dual Coxeter Number for Superalgberas

I am looking for a reference that gives the definition and has summarized the dual Coxeter number for superalgebras, especially for $\mathfrak{u}(m|n)$ (the Lie algebra of unitary supergroup $U(m|n)$)....

2
votes

0
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### Action of orthosymplectic group $SpO(d+1|d)$ on $PGL(d+1|d)$

Suppose $d$ is odd, and consider the super vector space $\mathbb{C}^{d+1|d}$ and its super projectivization $\mathbb{P}^{d|d}$.
In the case $d=1$, a paper of Witten's, concerning the moduli space of ...